src/HOL/Finite_Set.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51474 1e9e68247ad1
parent 51290 c48477e76de5
child 51487 f4bfdee99304
permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
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(*  Title:      HOL/Finite_Set.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Option Power
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begin
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subsection {* Predicate for finite sets *}
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inductive finite :: "'a set \<Rightarrow> bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
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simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
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lemma finite_induct [case_names empty insert, induct set: finite]:
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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  assumes "finite F"
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  assumes "P {}"
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    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
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  shows "P F"
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using `finite F`
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proof induct
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  show "P {}" by fact
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  fix x F assume F: "finite F" and P: "P F"
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  show "P (insert x F)"
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  proof cases
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    assume "x \<in> F"
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    hence "insert x F = F" by (rule insert_absorb)
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    with P show ?thesis by (simp only:)
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  next
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    assume "x \<notin> F"
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    from F this P show ?thesis by (rule insert)
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  qed
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qed
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subsubsection {* Choice principles *}
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from assms have "A \<noteq> UNIV" by blast
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  then show ?thesis by blast
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qed
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text {* A finite choice principle. Does not need the SOME choice operator. *}
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lemma finite_set_choice:
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  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
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proof (induct rule: finite_induct)
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  case empty then show ?case by simp
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next
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  case (insert a A)
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  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
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  show ?case (is "EX f. ?P f")
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  proof
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    show "?P(%x. if x = a then b else f x)" using f ab by auto
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  qed
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qed
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subsubsection {* Finite sets are the images of initial segments of natural numbers *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes "finite A" 
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  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
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using assms
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proof induct
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  case empty
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  show ?case
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  proof
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    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" by fact
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
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proof (induct n arbitrary: A)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
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  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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lemma finite_imp_inj_to_nat_seg:
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  assumes "finite A"
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  shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
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proof -
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  from finite_imp_nat_seg_image_inj_on[OF `finite A`]
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  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
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    by (auto simp:bij_betw_def)
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  let ?f = "the_inv_into {i. i<n} f"
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  have "inj_on ?f A & ?f ` A = {i. i<n}"
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    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
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  thus ?thesis by blast
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qed
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lemma finite_Collect_less_nat [iff]:
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  "finite {n::nat. n < k}"
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  by (fastforce simp: finite_conv_nat_seg_image)
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lemma finite_Collect_le_nat [iff]:
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  "finite {n::nat. n \<le> k}"
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  by (simp add: le_eq_less_or_eq Collect_disj_eq)
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subsubsection {* Finiteness and common set operations *}
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lemma rev_finite_subset:
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  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
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proof (induct arbitrary: A rule: finite_induct)
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  case empty
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  then show ?case by simp
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next
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  case (insert x F A)
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  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
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  show "finite A"
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  proof cases
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    assume x: "x \<in> A"
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    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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    with r have "finite (A - {x})" .
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    hence "finite (insert x (A - {x}))" ..
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    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
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    finally show ?thesis .
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  next
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    show "A \<subseteq> F ==> ?thesis" by fact
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    assume "x \<notin> A"
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    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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  qed
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qed
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lemma finite_subset:
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  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
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  by (rule rev_finite_subset)
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lemma finite_UnI:
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  assumes "finite F" and "finite G"
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  shows "finite (F \<union> G)"
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  using assms by induct simp_all
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lemma finite_Un [iff]:
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  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
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  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
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lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
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proof -
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  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
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  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
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  then show ?thesis by simp
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qed
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lemma finite_Int [simp, intro]:
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  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
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  by (blast intro: finite_subset)
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lemma finite_Collect_conjI [simp, intro]:
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  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
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  by (simp add: Collect_conj_eq)
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lemma finite_Collect_disjI [simp]:
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  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
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  by (simp add: Collect_disj_eq)
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lemma finite_Diff [simp, intro]:
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  "finite A \<Longrightarrow> finite (A - B)"
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  by (rule finite_subset, rule Diff_subset)
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lemma finite_Diff2 [simp]:
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  assumes "finite B"
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  shows "finite (A - B) \<longleftrightarrow> finite A"
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proof -
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  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
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  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
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  finally show ?thesis ..
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qed
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lemma finite_Diff_insert [iff]:
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  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
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proof -
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  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
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  moreover have "A - insert a B = A - B - {a}" by auto
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  ultimately show ?thesis by simp
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qed
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lemma finite_compl[simp]:
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  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Compl_eq_Diff_UNIV)
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lemma finite_Collect_not[simp]:
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  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Collect_neg_eq)
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lemma finite_Union [simp, intro]:
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  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN_I [intro]:
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  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN [simp]:
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  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
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  by (blast intro: finite_subset)
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lemma finite_Inter [intro]:
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  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
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  by (blast intro: Inter_lower finite_subset)
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lemma finite_INT [intro]:
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  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
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  by (blast intro: INT_lower finite_subset)
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lemma finite_imageI [simp, intro]:
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  "finite F \<Longrightarrow> finite (h ` F)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_image_set [simp]:
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  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
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  by (simp add: image_Collect [symmetric])
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lemma finite_imageD:
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  assumes "finite (f ` A)" and "inj_on f A"
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  shows "finite A"
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using assms
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proof (induct "f ` A" arbitrary: A)
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  case empty then show ?case by simp
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next
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  case (insert x B)
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  then have B_A: "insert x B = f ` A" by simp
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  then obtain y where "x = f y" and "y \<in> A" by blast
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  from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
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  with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
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  moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
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  ultimately have "finite (A - {y})" by (rule insert.hyps)
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  then show "finite A" by simp
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qed
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lemma finite_surj:
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  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
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  by (erule finite_subset) (rule finite_imageI)
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lemma finite_range_imageI:
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  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
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  by (drule finite_imageI) (simp add: range_composition)
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lemma finite_subset_image:
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  assumes "finite B"
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  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
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using assms
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proof induct
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  case empty then show ?case by simp
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next
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  case insert then show ?case
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    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
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       blast
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qed
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lemma finite_vimage_IntI:
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  "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
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  apply (induct rule: finite_induct)
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   apply simp_all
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  apply (subst vimage_insert)
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  apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
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  done
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lemma finite_vimageI:
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  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
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  using finite_vimage_IntI[of F h UNIV] by auto
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lemma finite_vimageD:
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  assumes fin: "finite (h -` F)" and surj: "surj h"
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  shows "finite F"
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proof -
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  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
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  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
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  finally show "finite F" .
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qed
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lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
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  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
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lemma finite_Collect_bex [simp]:
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  assumes "finite A"
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  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
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proof -
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  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
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  with assms show ?thesis by simp
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qed
wenzelm@12396
   320
haftmann@41656
   321
lemma finite_Collect_bounded_ex [simp]:
haftmann@41656
   322
  assumes "finite {y. P y}"
haftmann@41656
   323
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
haftmann@41656
   324
proof -
haftmann@41656
   325
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
haftmann@41656
   326
  with assms show ?thesis by simp
haftmann@41656
   327
qed
nipkow@29920
   328
haftmann@41656
   329
lemma finite_Plus:
haftmann@41656
   330
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
haftmann@41656
   331
  by (simp add: Plus_def)
nipkow@17022
   332
nipkow@31080
   333
lemma finite_PlusD: 
nipkow@31080
   334
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   335
  assumes fin: "finite (A <+> B)"
nipkow@31080
   336
  shows "finite A" "finite B"
nipkow@31080
   337
proof -
nipkow@31080
   338
  have "Inl ` A \<subseteq> A <+> B" by auto
haftmann@41656
   339
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   340
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   341
next
nipkow@31080
   342
  have "Inr ` B \<subseteq> A <+> B" by auto
haftmann@41656
   343
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   344
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   345
qed
nipkow@31080
   346
haftmann@41656
   347
lemma finite_Plus_iff [simp]:
haftmann@41656
   348
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
haftmann@41656
   349
  by (auto intro: finite_PlusD finite_Plus)
nipkow@31080
   350
haftmann@41656
   351
lemma finite_Plus_UNIV_iff [simp]:
haftmann@41656
   352
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
haftmann@41656
   353
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
wenzelm@12396
   354
nipkow@40786
   355
lemma finite_SigmaI [simp, intro]:
haftmann@41656
   356
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
nipkow@40786
   357
  by (unfold Sigma_def) blast
wenzelm@12396
   358
Andreas@51290
   359
lemma finite_SigmaI2:
Andreas@51290
   360
  assumes "finite {x\<in>A. B x \<noteq> {}}"
Andreas@51290
   361
  and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
Andreas@51290
   362
  shows "finite (Sigma A B)"
Andreas@51290
   363
proof -
Andreas@51290
   364
  from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
Andreas@51290
   365
  also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
Andreas@51290
   366
  finally show ?thesis .
Andreas@51290
   367
qed
Andreas@51290
   368
haftmann@41656
   369
lemma finite_cartesian_product:
haftmann@41656
   370
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   371
  by (rule finite_SigmaI)
nipkow@15402
   372
wenzelm@12396
   373
lemma finite_Prod_UNIV:
haftmann@41656
   374
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   375
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   376
paulson@15409
   377
lemma finite_cartesian_productD1:
haftmann@42207
   378
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
haftmann@42207
   379
  shows "finite A"
haftmann@42207
   380
proof -
haftmann@42207
   381
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   382
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   383
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
haftmann@42207
   384
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   385
    by (simp add: image_compose)
haftmann@42207
   386
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
haftmann@42207
   387
  then show ?thesis
haftmann@42207
   388
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   389
qed
paulson@15409
   390
paulson@15409
   391
lemma finite_cartesian_productD2:
haftmann@42207
   392
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
haftmann@42207
   393
  shows "finite B"
haftmann@42207
   394
proof -
haftmann@42207
   395
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   396
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   397
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
haftmann@42207
   398
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   399
    by (simp add: image_compose)
haftmann@42207
   400
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
haftmann@42207
   401
  then show ?thesis
haftmann@42207
   402
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   403
qed
paulson@15409
   404
Andreas@48175
   405
lemma finite_prod: 
Andreas@48175
   406
  "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
Andreas@48175
   407
by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV 
Andreas@48175
   408
   dest: finite_cartesian_productD1 finite_cartesian_productD2)
Andreas@48175
   409
haftmann@41656
   410
lemma finite_Pow_iff [iff]:
haftmann@41656
   411
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   412
proof
wenzelm@12396
   413
  assume "finite (Pow A)"
haftmann@41656
   414
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   415
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   416
next
wenzelm@12396
   417
  assume "finite A"
haftmann@41656
   418
  then show "finite (Pow A)"
huffman@35216
   419
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   420
qed
wenzelm@12396
   421
haftmann@41656
   422
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   423
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   424
  by (simp add: Pow_def [symmetric])
nipkow@29918
   425
Andreas@48175
   426
lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
Andreas@48175
   427
by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
Andreas@48175
   428
nipkow@15392
   429
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   430
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   431
nipkow@15392
   432
haftmann@41656
   433
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   434
haftmann@41656
   435
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   436
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   437
  assumes "\<And>x. P {x}"
haftmann@41656
   438
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   439
  shows "P F"
wenzelm@46898
   440
using assms
wenzelm@46898
   441
proof induct
haftmann@41656
   442
  case empty then show ?case by simp
haftmann@41656
   443
next
haftmann@41656
   444
  case (insert x F) then show ?case by cases auto
haftmann@41656
   445
qed
haftmann@41656
   446
haftmann@41656
   447
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   448
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   449
  assumes empty: "P {}"
haftmann@41656
   450
    and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
haftmann@41656
   451
  shows "P F"
wenzelm@46898
   452
using `finite F` `F \<subseteq> A`
wenzelm@46898
   453
proof induct
haftmann@41656
   454
  show "P {}" by fact
nipkow@31441
   455
next
haftmann@41656
   456
  fix x F
haftmann@41656
   457
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   458
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   459
  show "P (insert x F)"
haftmann@41656
   460
  proof (rule insert)
haftmann@41656
   461
    from i show "x \<in> A" by blast
haftmann@41656
   462
    from i have "F \<subseteq> A" by blast
haftmann@41656
   463
    with P show "P F" .
haftmann@41656
   464
    show "finite F" by fact
haftmann@41656
   465
    show "x \<notin> F" by fact
haftmann@41656
   466
  qed
haftmann@41656
   467
qed
haftmann@41656
   468
haftmann@41656
   469
lemma finite_empty_induct:
haftmann@41656
   470
  assumes "finite A"
haftmann@41656
   471
  assumes "P A"
haftmann@41656
   472
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   473
  shows "P {}"
haftmann@41656
   474
proof -
haftmann@41656
   475
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   476
  proof -
haftmann@41656
   477
    fix B :: "'a set"
haftmann@41656
   478
    assume "B \<subseteq> A"
haftmann@41656
   479
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   480
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   481
    proof induct
haftmann@41656
   482
      case empty
haftmann@41656
   483
      from `P A` show ?case by simp
haftmann@41656
   484
    next
haftmann@41656
   485
      case (insert b B)
haftmann@41656
   486
      have "P (A - B - {b})"
haftmann@41656
   487
      proof (rule remove)
haftmann@41656
   488
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   489
        from insert show "b \<in> A - B" by simp
haftmann@41656
   490
        from insert show "P (A - B)" by simp
haftmann@41656
   491
      qed
haftmann@41656
   492
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   493
      finally show ?case .
haftmann@41656
   494
    qed
haftmann@41656
   495
  qed
haftmann@41656
   496
  then have "P (A - A)" by blast
haftmann@41656
   497
  then show ?thesis by simp
nipkow@31441
   498
qed
nipkow@31441
   499
nipkow@31441
   500
haftmann@26441
   501
subsection {* Class @{text finite}  *}
haftmann@26041
   502
haftmann@29797
   503
class finite =
haftmann@26041
   504
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   505
begin
huffman@27430
   506
huffman@27430
   507
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   508
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   509
haftmann@43866
   510
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
bulwahn@40922
   511
  by simp
bulwahn@40922
   512
huffman@27430
   513
end
huffman@27430
   514
wenzelm@46898
   515
instance prod :: (finite, finite) finite
wenzelm@46898
   516
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   517
haftmann@26041
   518
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   519
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   520
haftmann@26146
   521
instance "fun" :: (finite, finite) finite
haftmann@26146
   522
proof
haftmann@26041
   523
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   524
  proof (rule finite_imageD)
haftmann@26041
   525
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   526
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   527
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   528
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   529
    ultimately show "finite (range ?graph)"
berghofe@26792
   530
      by (rule finite_subset)
haftmann@26041
   531
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   532
  qed
haftmann@26041
   533
qed
haftmann@26041
   534
wenzelm@46898
   535
instance bool :: finite
wenzelm@46898
   536
  by default (simp add: UNIV_bool)
haftmann@44831
   537
haftmann@45962
   538
instance set :: (finite) finite
haftmann@45962
   539
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
haftmann@45962
   540
wenzelm@46898
   541
instance unit :: finite
wenzelm@46898
   542
  by default (simp add: UNIV_unit)
haftmann@44831
   543
wenzelm@46898
   544
instance sum :: (finite, finite) finite
wenzelm@46898
   545
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   546
haftmann@44831
   547
lemma finite_option_UNIV [simp]:
haftmann@44831
   548
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@44831
   549
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@44831
   550
wenzelm@46898
   551
instance option :: (finite) finite
wenzelm@46898
   552
  by default (simp add: UNIV_option_conv)
haftmann@44831
   553
haftmann@26041
   554
haftmann@35817
   555
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   556
nipkow@15392
   557
text {* The intended behaviour is
wenzelm@31916
   558
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   559
if @{text f} is ``left-commutative'':
nipkow@15392
   560
*}
nipkow@15392
   561
haftmann@42871
   562
locale comp_fun_commute =
nipkow@28853
   563
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
   564
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
nipkow@28853
   565
begin
nipkow@28853
   566
haftmann@42809
   567
lemma fun_left_comm: "f x (f y z) = f y (f x z)"
haftmann@42871
   568
  using comp_fun_commute by (simp add: fun_eq_iff)
nipkow@28853
   569
nipkow@28853
   570
end
nipkow@28853
   571
nipkow@28853
   572
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   573
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   574
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   575
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   576
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   577
nipkow@28853
   578
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   579
nipkow@28853
   580
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   581
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   582
paulson@15498
   583
text{*A tempting alternative for the definiens is
nipkow@28853
   584
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   585
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   586
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   587
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   588
nipkow@28853
   589
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   590
by (induct rule: finite_induct) auto
nipkow@28853
   591
nipkow@28853
   592
nipkow@28853
   593
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   594
haftmann@42871
   595
context comp_fun_commute
haftmann@26041
   596
begin
haftmann@26041
   597
huffman@36045
   598
lemma fold_graph_insertE_aux:
huffman@36045
   599
  "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
huffman@36045
   600
proof (induct set: fold_graph)
huffman@36045
   601
  case (insertI x A y) show ?case
huffman@36045
   602
  proof (cases "x = a")
huffman@36045
   603
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   604
  next
huffman@36045
   605
    assume "x \<noteq> a"
huffman@36045
   606
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   607
      using insertI by auto
haftmann@42875
   608
    have "f x y = f a (f x y')"
huffman@36045
   609
      unfolding y by (rule fun_left_comm)
haftmann@42875
   610
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   611
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   612
      by (simp add: insert_Diff_if fold_graph.insertI)
haftmann@42875
   613
    ultimately show ?case by fast
nipkow@15392
   614
  qed
huffman@36045
   615
qed simp
huffman@36045
   616
huffman@36045
   617
lemma fold_graph_insertE:
huffman@36045
   618
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   619
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   620
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   621
nipkow@28853
   622
lemma fold_graph_determ:
nipkow@28853
   623
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   624
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   625
  case (insertI x A y v)
huffman@36045
   626
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   627
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   628
    by (rule fold_graph_insertE)
huffman@36045
   629
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   630
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   631
qed fast
nipkow@15392
   632
nipkow@28853
   633
lemma fold_equality:
nipkow@28853
   634
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   635
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   636
haftmann@42272
   637
lemma fold_graph_fold:
haftmann@42272
   638
  assumes "finite A"
haftmann@42272
   639
  shows "fold_graph f z A (fold f z A)"
haftmann@42272
   640
proof -
haftmann@42272
   641
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
haftmann@42272
   642
  moreover note fold_graph_determ
haftmann@42272
   643
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
haftmann@42272
   644
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
haftmann@42272
   645
  then show ?thesis by (unfold fold_def)
haftmann@42272
   646
qed
huffman@36045
   647
nipkow@15392
   648
text{* The base case for @{text fold}: *}
nipkow@15392
   649
nipkow@28853
   650
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   651
by (unfold fold_def) blast
nipkow@28853
   652
nipkow@28853
   653
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   654
haftmann@26041
   655
lemma fold_insert [simp]:
haftmann@42875
   656
  assumes "finite A" and "x \<notin> A"
haftmann@42875
   657
  shows "fold f z (insert x A) = f x (fold f z A)"
haftmann@42875
   658
proof (rule fold_equality)
haftmann@42875
   659
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
haftmann@42875
   660
  with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
haftmann@42875
   661
qed
nipkow@28853
   662
nipkow@28853
   663
lemma fold_fun_comm:
nipkow@28853
   664
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   665
proof (induct rule: finite_induct)
nipkow@28853
   666
  case empty then show ?case by simp
nipkow@28853
   667
next
nipkow@28853
   668
  case (insert y A) then show ?case
nipkow@28853
   669
    by (simp add: fun_left_comm[of x])
nipkow@28853
   670
qed
nipkow@28853
   671
nipkow@28853
   672
lemma fold_insert2:
nipkow@28853
   673
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   674
by (simp add: fold_fun_comm)
nipkow@15392
   675
haftmann@26041
   676
lemma fold_rec:
haftmann@42875
   677
  assumes "finite A" and "x \<in> A"
haftmann@42875
   678
  shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   679
proof -
nipkow@28853
   680
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   681
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   682
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   683
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   684
  finally show ?thesis .
nipkow@15535
   685
qed
nipkow@15535
   686
nipkow@28853
   687
lemma fold_insert_remove:
nipkow@28853
   688
  assumes "finite A"
nipkow@28853
   689
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   690
proof -
nipkow@28853
   691
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   692
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   693
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   694
    by (rule fold_rec)
nipkow@28853
   695
  then show ?thesis by simp
nipkow@28853
   696
qed
nipkow@28853
   697
kuncar@48619
   698
text{* Other properties of @{const fold}: *}
kuncar@48619
   699
kuncar@48619
   700
lemma fold_image:
kuncar@48619
   701
  assumes "finite A" and "inj_on g A"
kuncar@48619
   702
  shows "fold f x (g ` A) = fold (f \<circ> g) x A"
kuncar@48619
   703
using assms
kuncar@48619
   704
proof induction
kuncar@48619
   705
  case (insert a F)
kuncar@48619
   706
    interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
kuncar@48619
   707
    from insert show ?case by auto
kuncar@48619
   708
qed (simp)
kuncar@48619
   709
haftmann@26041
   710
end
nipkow@15392
   711
haftmann@49724
   712
lemma fold_cong:
haftmann@49724
   713
  assumes "comp_fun_commute f" "comp_fun_commute g"
haftmann@49724
   714
  assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
haftmann@49724
   715
    and "A = B" and "s = t"
haftmann@49724
   716
  shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
haftmann@49724
   717
proof -
haftmann@49724
   718
  have "Finite_Set.fold f s A = Finite_Set.fold g s A"  
haftmann@49724
   719
  using `finite A` cong proof (induct A)
haftmann@49724
   720
    case empty then show ?case by simp
haftmann@49724
   721
  next
haftmann@49724
   722
    case (insert x A)
haftmann@49724
   723
    interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
haftmann@49724
   724
    interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
haftmann@49724
   725
    from insert show ?case by simp
haftmann@49724
   726
  qed
haftmann@49724
   727
  with assms show ?thesis by simp
haftmann@49724
   728
qed
haftmann@49724
   729
haftmann@49724
   730
nipkow@15480
   731
text{* A simplified version for idempotent functions: *}
nipkow@15480
   732
haftmann@42871
   733
locale comp_fun_idem = comp_fun_commute +
haftmann@42871
   734
  assumes comp_fun_idem: "f x o f x = f x"
haftmann@26041
   735
begin
haftmann@26041
   736
haftmann@42869
   737
lemma fun_left_idem: "f x (f x z) = f x z"
haftmann@42871
   738
  using comp_fun_idem by (simp add: fun_eq_iff)
nipkow@28853
   739
haftmann@26041
   740
lemma fold_insert_idem:
nipkow@28853
   741
  assumes fin: "finite A"
nipkow@28853
   742
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   743
proof cases
nipkow@28853
   744
  assume "x \<in> A"
nipkow@28853
   745
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   746
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   747
next
nipkow@28853
   748
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   749
qed
nipkow@15480
   750
nipkow@28853
   751
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   752
nipkow@28853
   753
lemma fold_insert_idem2:
nipkow@28853
   754
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   755
by(simp add:fold_fun_comm)
nipkow@15484
   756
haftmann@26041
   757
end
haftmann@26041
   758
haftmann@35817
   759
haftmann@49723
   760
subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
haftmann@35817
   761
haftmann@42871
   762
lemma (in comp_fun_commute) comp_comp_fun_commute:
haftmann@42871
   763
  "comp_fun_commute (f \<circ> g)"
haftmann@35817
   764
proof
haftmann@42871
   765
qed (simp_all add: comp_fun_commute)
haftmann@35817
   766
haftmann@42871
   767
lemma (in comp_fun_idem) comp_comp_fun_idem:
haftmann@42871
   768
  "comp_fun_idem (f \<circ> g)"
haftmann@42871
   769
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
haftmann@42871
   770
    (simp_all add: comp_fun_idem)
haftmann@35817
   771
haftmann@49723
   772
lemma (in comp_fun_commute) comp_fun_commute_funpow:
haftmann@49723
   773
  "comp_fun_commute (\<lambda>x. f x ^^ g x)"
haftmann@49723
   774
proof
haftmann@49723
   775
  fix y x
haftmann@49723
   776
  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
haftmann@49723
   777
  proof (cases "x = y")
haftmann@49723
   778
    case True then show ?thesis by simp
haftmann@49723
   779
  next
haftmann@49723
   780
    case False show ?thesis
haftmann@49723
   781
    proof (induct "g x" arbitrary: g)
haftmann@49723
   782
      case 0 then show ?case by simp
haftmann@49723
   783
    next
haftmann@49723
   784
      case (Suc n g)
haftmann@49723
   785
      have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
haftmann@49723
   786
      proof (induct "g y" arbitrary: g)
haftmann@49723
   787
        case 0 then show ?case by simp
haftmann@49723
   788
      next
haftmann@49723
   789
        case (Suc n g)
haftmann@49723
   790
        def h \<equiv> "\<lambda>z. g z - 1"
haftmann@49723
   791
        with Suc have "n = h y" by simp
haftmann@49723
   792
        with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
haftmann@49723
   793
          by auto
haftmann@49723
   794
        from Suc h_def have "g y = Suc (h y)" by simp
haftmann@49739
   795
        then show ?case by (simp add: comp_assoc hyp)
haftmann@49723
   796
          (simp add: o_assoc comp_fun_commute)
haftmann@49723
   797
      qed
haftmann@49723
   798
      def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
haftmann@49723
   799
      with Suc have "n = h x" by simp
haftmann@49723
   800
      with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
haftmann@49723
   801
        by auto
haftmann@49723
   802
      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
haftmann@49723
   803
      from Suc h_def have "g x = Suc (h x)" by simp
haftmann@49723
   804
      then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
haftmann@49739
   805
        (simp add: comp_assoc hyp1)
haftmann@49723
   806
    qed
haftmann@49723
   807
  qed
haftmann@49723
   808
qed
haftmann@49723
   809
haftmann@49723
   810
haftmann@49723
   811
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@49723
   812
haftmann@42871
   813
lemma comp_fun_idem_insert:
haftmann@42871
   814
  "comp_fun_idem insert"
haftmann@35817
   815
proof
haftmann@35817
   816
qed auto
haftmann@35817
   817
haftmann@42871
   818
lemma comp_fun_idem_remove:
haftmann@46146
   819
  "comp_fun_idem Set.remove"
haftmann@35817
   820
proof
haftmann@35817
   821
qed auto
nipkow@31992
   822
haftmann@42871
   823
lemma (in semilattice_inf) comp_fun_idem_inf:
haftmann@42871
   824
  "comp_fun_idem inf"
haftmann@35817
   825
proof
haftmann@35817
   826
qed (auto simp add: inf_left_commute)
haftmann@35817
   827
haftmann@42871
   828
lemma (in semilattice_sup) comp_fun_idem_sup:
haftmann@42871
   829
  "comp_fun_idem sup"
haftmann@35817
   830
proof
haftmann@35817
   831
qed (auto simp add: sup_left_commute)
nipkow@31992
   832
haftmann@35817
   833
lemma union_fold_insert:
haftmann@35817
   834
  assumes "finite A"
haftmann@35817
   835
  shows "A \<union> B = fold insert B A"
haftmann@35817
   836
proof -
haftmann@42871
   837
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
haftmann@35817
   838
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   839
qed
nipkow@31992
   840
haftmann@35817
   841
lemma minus_fold_remove:
haftmann@35817
   842
  assumes "finite A"
haftmann@46146
   843
  shows "B - A = fold Set.remove B A"
haftmann@35817
   844
proof -
haftmann@46146
   845
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
haftmann@46146
   846
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
haftmann@46146
   847
  then show ?thesis ..
haftmann@35817
   848
qed
haftmann@35817
   849
kuncar@48619
   850
lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
kuncar@48619
   851
proof - 
kuncar@48619
   852
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
kuncar@48619
   853
  show ?thesis by default (auto simp: fun_eq_iff)
kuncar@48619
   854
qed
kuncar@48619
   855
kuncar@49758
   856
lemma Set_filter_fold:
kuncar@48619
   857
  assumes "finite A"
kuncar@49758
   858
  shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
kuncar@48619
   859
using assms
kuncar@48619
   860
by (induct A) 
kuncar@49758
   861
  (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
kuncar@49758
   862
kuncar@49758
   863
lemma inter_Set_filter:     
kuncar@49758
   864
  assumes "finite B"
kuncar@49758
   865
  shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
kuncar@49758
   866
using assms 
kuncar@49758
   867
by (induct B) (auto simp: Set.filter_def)
kuncar@48619
   868
kuncar@48619
   869
lemma image_fold_insert:
kuncar@48619
   870
  assumes "finite A"
kuncar@48619
   871
  shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
kuncar@48619
   872
using assms
kuncar@48619
   873
proof -
kuncar@48619
   874
  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
kuncar@48619
   875
  show ?thesis using assms by (induct A) auto
kuncar@48619
   876
qed
kuncar@48619
   877
kuncar@48619
   878
lemma Ball_fold:
kuncar@48619
   879
  assumes "finite A"
kuncar@48619
   880
  shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
kuncar@48619
   881
using assms
kuncar@48619
   882
proof -
kuncar@48619
   883
  interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
kuncar@48619
   884
  show ?thesis using assms by (induct A) auto
kuncar@48619
   885
qed
kuncar@48619
   886
kuncar@48619
   887
lemma Bex_fold:
kuncar@48619
   888
  assumes "finite A"
kuncar@48619
   889
  shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
kuncar@48619
   890
using assms
kuncar@48619
   891
proof -
kuncar@48619
   892
  interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
kuncar@48619
   893
  show ?thesis using assms by (induct A) auto
kuncar@48619
   894
qed
kuncar@48619
   895
kuncar@48619
   896
lemma comp_fun_commute_Pow_fold: 
kuncar@48619
   897
  "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
kuncar@48619
   898
  by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
kuncar@48619
   899
kuncar@48619
   900
lemma Pow_fold:
kuncar@48619
   901
  assumes "finite A"
kuncar@48619
   902
  shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
kuncar@48619
   903
using assms
kuncar@48619
   904
proof -
kuncar@48619
   905
  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
kuncar@48619
   906
  show ?thesis using assms by (induct A) (auto simp: Pow_insert)
kuncar@48619
   907
qed
kuncar@48619
   908
kuncar@48619
   909
lemma fold_union_pair:
kuncar@48619
   910
  assumes "finite B"
kuncar@48619
   911
  shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
kuncar@48619
   912
proof -
kuncar@48619
   913
  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
kuncar@48619
   914
  show ?thesis using assms  by (induct B arbitrary: A) simp_all
kuncar@48619
   915
qed
kuncar@48619
   916
kuncar@48619
   917
lemma comp_fun_commute_product_fold: 
kuncar@48619
   918
  assumes "finite B"
kuncar@48619
   919
  shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)" 
kuncar@48619
   920
by default (auto simp: fold_union_pair[symmetric] assms)
kuncar@48619
   921
kuncar@48619
   922
lemma product_fold:
kuncar@48619
   923
  assumes "finite A"
kuncar@48619
   924
  assumes "finite B"
kuncar@48619
   925
  shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
kuncar@48619
   926
using assms unfolding Sigma_def 
kuncar@48619
   927
by (induct A) 
kuncar@48619
   928
  (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
kuncar@48619
   929
kuncar@48619
   930
haftmann@35817
   931
context complete_lattice
nipkow@31992
   932
begin
nipkow@31992
   933
haftmann@35817
   934
lemma inf_Inf_fold_inf:
haftmann@35817
   935
  assumes "finite A"
haftmann@35817
   936
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   937
proof -
haftmann@42871
   938
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@35817
   939
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   940
    (simp_all add: inf_commute fold_fun_comm)
haftmann@35817
   941
qed
nipkow@31992
   942
haftmann@35817
   943
lemma sup_Sup_fold_sup:
haftmann@35817
   944
  assumes "finite A"
haftmann@35817
   945
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   946
proof -
haftmann@42871
   947
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@35817
   948
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   949
    (simp_all add: sup_commute fold_fun_comm)
nipkow@31992
   950
qed
nipkow@31992
   951
haftmann@35817
   952
lemma Inf_fold_inf:
haftmann@35817
   953
  assumes "finite A"
haftmann@35817
   954
  shows "Inf A = fold inf top A"
haftmann@35817
   955
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   956
haftmann@35817
   957
lemma Sup_fold_sup:
haftmann@35817
   958
  assumes "finite A"
haftmann@35817
   959
  shows "Sup A = fold sup bot A"
haftmann@35817
   960
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   961
haftmann@46146
   962
lemma inf_INF_fold_inf:
haftmann@35817
   963
  assumes "finite A"
haftmann@42873
   964
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
haftmann@35817
   965
proof (rule sym)
haftmann@42871
   966
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@42871
   967
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   968
  from `finite A` show "?fold = ?inf"
haftmann@42869
   969
    by (induct A arbitrary: B)
hoelzl@44928
   970
      (simp_all add: INF_def inf_left_commute)
haftmann@35817
   971
qed
nipkow@31992
   972
haftmann@46146
   973
lemma sup_SUP_fold_sup:
haftmann@35817
   974
  assumes "finite A"
haftmann@42873
   975
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
haftmann@35817
   976
proof (rule sym)
haftmann@42871
   977
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@42871
   978
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   979
  from `finite A` show "?fold = ?sup"
haftmann@42869
   980
    by (induct A arbitrary: B)
hoelzl@44928
   981
      (simp_all add: SUP_def sup_left_commute)
haftmann@35817
   982
qed
nipkow@31992
   983
haftmann@46146
   984
lemma INF_fold_inf:
haftmann@35817
   985
  assumes "finite A"
haftmann@42873
   986
  shows "INFI A f = fold (inf \<circ> f) top A"
haftmann@46146
   987
  using assms inf_INF_fold_inf [of A top] by simp
nipkow@31992
   988
haftmann@46146
   989
lemma SUP_fold_sup:
haftmann@35817
   990
  assumes "finite A"
haftmann@42873
   991
  shows "SUPR A f = fold (sup \<circ> f) bot A"
haftmann@46146
   992
  using assms sup_SUP_fold_sup [of A bot] by simp
nipkow@31992
   993
nipkow@31992
   994
end
nipkow@31992
   995
nipkow@31992
   996
haftmann@35817
   997
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   998
nipkow@28853
   999
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
haftmann@42875
  1000
  where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
nipkow@28853
  1001
nipkow@28853
  1002
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
haftmann@42875
  1003
  by (simp add:fold_image_def)
nipkow@15392
  1004
haftmann@26041
  1005
context ab_semigroup_mult
haftmann@26041
  1006
begin
haftmann@26041
  1007
nipkow@28853
  1008
lemma fold_image_insert[simp]:
haftmann@42875
  1009
  assumes "finite A" and "a \<notin> A"
haftmann@42875
  1010
  shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
  1011
proof -
wenzelm@46898
  1012
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
  1013
    by default (simp add: fun_eq_iff mult_ac)
wenzelm@46898
  1014
  from assms show ?thesis by (simp add: fold_image_def)
nipkow@28853
  1015
qed
nipkow@28853
  1016
nipkow@28853
  1017
lemma fold_image_reindex:
haftmann@42875
  1018
  assumes "finite A"
haftmann@42875
  1019
  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
haftmann@42875
  1020
  using assms by induct auto
nipkow@28853
  1021
nipkow@28853
  1022
lemma fold_image_cong:
haftmann@42875
  1023
  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
haftmann@42875
  1024
  shows "fold_image times g z A = fold_image times h z A"
haftmann@42875
  1025
proof -
haftmann@42875
  1026
  from `finite A`
haftmann@42875
  1027
  have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
haftmann@42875
  1028
  proof (induct arbitrary: C)
haftmann@42875
  1029
    case empty then show ?case by simp
haftmann@42875
  1030
  next
haftmann@42875
  1031
    case (insert x F) then show ?case apply -
haftmann@42875
  1032
    apply (simp add: subset_insert_iff, clarify)
haftmann@42875
  1033
    apply (subgoal_tac "finite C")
wenzelm@48125
  1034
      prefer 2 apply (blast dest: finite_subset [rotated])
haftmann@42875
  1035
    apply (subgoal_tac "C = insert x (C - {x})")
haftmann@42875
  1036
      prefer 2 apply blast
haftmann@42875
  1037
    apply (erule ssubst)
haftmann@42875
  1038
    apply (simp add: Ball_def del: insert_Diff_single)
haftmann@42875
  1039
    done
haftmann@42875
  1040
  qed
haftmann@42875
  1041
  with g_h show ?thesis by simp
haftmann@42875
  1042
qed
nipkow@15392
  1043
haftmann@26041
  1044
end
haftmann@26041
  1045
haftmann@26041
  1046
context comm_monoid_mult
haftmann@26041
  1047
begin
haftmann@26041
  1048
haftmann@35817
  1049
lemma fold_image_1:
haftmann@35817
  1050
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@41656
  1051
  apply (induct rule: finite_induct)
haftmann@35817
  1052
  apply simp by auto
haftmann@35817
  1053
nipkow@28853
  1054
lemma fold_image_Un_Int:
haftmann@26041
  1055
  "finite A ==> finite B ==>
nipkow@28853
  1056
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
  1057
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
haftmann@41656
  1058
  apply (induct rule: finite_induct)
nipkow@28853
  1059
by (induct set: finite) 
nipkow@28853
  1060
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
  1061
haftmann@35817
  1062
lemma fold_image_Un_one:
haftmann@35817
  1063
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
  1064
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
  1065
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
  1066
proof-
haftmann@35817
  1067
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
  1068
    apply (rule fold_image_1)
haftmann@35817
  1069
    using fS fT I0 by auto 
haftmann@35817
  1070
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
  1071
qed
haftmann@35817
  1072
haftmann@26041
  1073
corollary fold_Un_disjoint:
haftmann@26041
  1074
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
  1075
   fold_image times g 1 (A Un B) =
nipkow@28853
  1076
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
  1077
by (simp add: fold_image_Un_Int)
nipkow@28853
  1078
nipkow@28853
  1079
lemma fold_image_UN_disjoint:
haftmann@26041
  1080
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
  1081
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
  1082
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
  1083
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
haftmann@41656
  1084
apply (induct rule: finite_induct)
haftmann@41656
  1085
apply simp
haftmann@41656
  1086
apply atomize
nipkow@28853
  1087
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
  1088
 prefer 2 apply blast
nipkow@28853
  1089
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
  1090
 prefer 2 apply blast
nipkow@28853
  1091
apply (simp add: fold_Un_disjoint)
nipkow@28853
  1092
done
nipkow@28853
  1093
nipkow@28853
  1094
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
  1095
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
  1096
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
  1097
apply (subst Sigma_def)
nipkow@28853
  1098
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
  1099
 apply blast
nipkow@28853
  1100
apply (erule fold_image_cong)
nipkow@28853
  1101
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
  1102
 apply blast
paulson@15506
  1103
apply simp
nipkow@15392
  1104
done
nipkow@15392
  1105
nipkow@28853
  1106
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
  1107
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
  1108
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
  1109
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
  1110
chaieb@30260
  1111
lemma fold_image_related: 
chaieb@30260
  1112
  assumes Re: "R e e" 
chaieb@30260
  1113
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
  1114
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
  1115
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
  1116
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
  1117
chaieb@30260
  1118
lemma  fold_image_eq_general:
chaieb@30260
  1119
  assumes fS: "finite S"
chaieb@30260
  1120
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
  1121
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
  1122
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
  1123
proof-
chaieb@30260
  1124
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
  1125
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
  1126
    from f12 h H  have "x = y" by auto }
chaieb@30260
  1127
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
  1128
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
  1129
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
  1130
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
  1131
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
  1132
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
  1133
    by blast
chaieb@30260
  1134
  finally show ?thesis ..
chaieb@30260
  1135
qed
chaieb@30260
  1136
chaieb@30260
  1137
lemma fold_image_eq_general_inverses:
chaieb@30260
  1138
  assumes fS: "finite S" 
chaieb@30260
  1139
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
  1140
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
chaieb@30260
  1141
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
  1142
  (* metis solves it, but not yet available here *)
chaieb@30260
  1143
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
  1144
  apply (rule ballI)
chaieb@30260
  1145
  apply (frule kh)
chaieb@30260
  1146
  apply (rule ex1I[])
chaieb@30260
  1147
  apply blast
chaieb@30260
  1148
  apply clarsimp
chaieb@30260
  1149
  apply (drule hk) apply simp
chaieb@30260
  1150
  apply (rule sym)
chaieb@30260
  1151
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
  1152
  apply (rule ballI)
chaieb@30260
  1153
  apply (drule  hk)
chaieb@30260
  1154
  apply blast
chaieb@30260
  1155
  done
chaieb@30260
  1156
haftmann@26041
  1157
end
haftmann@22917
  1158
nipkow@25162
  1159
haftmann@35817
  1160
subsection {* A fold functional for non-empty sets *}
nipkow@15392
  1161
nipkow@15392
  1162
text{* Does not require start value. *}
wenzelm@12396
  1163
berghofe@23736
  1164
inductive
berghofe@22262
  1165
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1166
  for f :: "'a => 'a => 'a"
berghofe@22262
  1167
where
paulson@15506
  1168
  fold1Set_insertI [intro]:
nipkow@28853
  1169
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1170
haftmann@35416
  1171
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
  1172
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1173
paulson@15506
  1174
lemma fold1Set_nonempty:
haftmann@22917
  1175
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1176
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1177
berghofe@23736
  1178
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1179
berghofe@23736
  1180
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1181
berghofe@22262
  1182
berghofe@22262
  1183
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1184
by (blast elim: fold_graph.cases)
nipkow@15392
  1185
haftmann@22917
  1186
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1187
by (unfold fold1_def) blast
wenzelm@12396
  1188
paulson@15508
  1189
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1190
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1191
apply (induct A rule: finite_induct)
nipkow@28853
  1192
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1193
done
paulson@15506
  1194
nipkow@28853
  1195
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1196
haftmann@26041
  1197
context ab_semigroup_mult
haftmann@26041
  1198
begin
haftmann@26041
  1199
wenzelm@46898
  1200
lemma comp_fun_commute: "comp_fun_commute (op *)"
wenzelm@46898
  1201
  by default (simp add: fun_eq_iff mult_ac)
nipkow@28853
  1202
nipkow@28853
  1203
lemma fold_graph_insert_swap:
nipkow@28853
  1204
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1205
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1206
proof -
haftmann@42871
  1207
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1208
from assms show ?thesis
nipkow@28853
  1209
proof (induct rule: fold_graph.induct)
huffman@36045
  1210
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1211
next
berghofe@22262
  1212
  case (insertI x A y)
nipkow@28853
  1213
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1214
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1215
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1216
qed
nipkow@28853
  1217
qed
nipkow@28853
  1218
nipkow@28853
  1219
lemma fold_graph_permute_diff:
nipkow@28853
  1220
assumes fold: "fold_graph times b A x"
nipkow@28853
  1221
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1222
using fold
nipkow@28853
  1223
proof (induct rule: fold_graph.induct)
paulson@15508
  1224
  case emptyI thus ?case by simp
paulson@15508
  1225
next
berghofe@22262
  1226
  case (insertI x A y)
paulson@15521
  1227
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1228
  thus ?case
paulson@15521
  1229
  proof
paulson@15521
  1230
    assume "a = x"
paulson@15521
  1231
    with insertI show ?thesis
nipkow@28853
  1232
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1233
  next
paulson@15521
  1234
    assume ainA: "a \<in> A"
nipkow@28853
  1235
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1236
      using insertI by force
paulson@15521
  1237
    moreover
paulson@15521
  1238
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1239
      using ainA insertI by blast
nipkow@28853
  1240
    ultimately show ?thesis by simp
paulson@15508
  1241
  qed
paulson@15508
  1242
qed
paulson@15508
  1243
haftmann@26041
  1244
lemma fold1_eq_fold:
nipkow@28853
  1245
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1246
proof -
haftmann@42871
  1247
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1248
  from assms show ?thesis
nipkow@28853
  1249
apply (simp add: fold1_def fold_def)
paulson@15508
  1250
apply (rule the_equality)
nipkow@28853
  1251
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1252
apply (rule sym, clarify)
paulson@15508
  1253
apply (case_tac "Aa=A")
huffman@35216
  1254
 apply (best intro: fold_graph_determ)
nipkow@28853
  1255
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1256
 apply (best intro: fold_graph_determ)
nipkow@28853
  1257
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1258
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1259
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1260
done
nipkow@28853
  1261
qed
paulson@15508
  1262
paulson@15521
  1263
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1264
apply safe
nipkow@28853
  1265
 apply simp
nipkow@28853
  1266
 apply (drule_tac x=x in spec)
nipkow@28853
  1267
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1268
done
paulson@15508
  1269
haftmann@26041
  1270
lemma fold1_insert:
paulson@15521
  1271
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1272
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1273
proof -
haftmann@42871
  1274
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1275
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1276
    by (auto simp add: nonempty_iff)
paulson@15521
  1277
  with A show ?thesis
nipkow@28853
  1278
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1279
qed
paulson@15521
  1280
haftmann@26041
  1281
end
haftmann@26041
  1282
haftmann@26041
  1283
context ab_semigroup_idem_mult
haftmann@26041
  1284
begin
haftmann@26041
  1285
wenzelm@46898
  1286
lemma comp_fun_idem: "comp_fun_idem (op *)"
wenzelm@46898
  1287
  by default (simp_all add: fun_eq_iff mult_left_commute)
haftmann@35817
  1288
haftmann@26041
  1289
lemma fold1_insert_idem [simp]:
paulson@15521
  1290
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1291
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1292
proof -
haftmann@42871
  1293
  interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@42871
  1294
    by (rule comp_fun_idem)
nipkow@28853
  1295
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1296
    by (auto simp add: nonempty_iff)
paulson@15521
  1297
  show ?thesis
paulson@15521
  1298
  proof cases
wenzelm@41550
  1299
    assume a: "a = x"
wenzelm@41550
  1300
    show ?thesis
paulson@15521
  1301
    proof cases
paulson@15521
  1302
      assume "A' = {}"
wenzelm@41550
  1303
      with A' a show ?thesis by simp
paulson@15521
  1304
    next
paulson@15521
  1305
      assume "A' \<noteq> {}"
wenzelm@41550
  1306
      with A A' a show ?thesis
huffman@35216
  1307
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1308
    qed
paulson@15521
  1309
  next
paulson@15521
  1310
    assume "a \<noteq> x"
wenzelm@41550
  1311
    with A A' show ?thesis
huffman@35216
  1312
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1313
  qed
paulson@15521
  1314
qed
paulson@15506
  1315
haftmann@26041
  1316
lemma hom_fold1_commute:
haftmann@26041
  1317
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1318
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
wenzelm@46898
  1319
using N
wenzelm@46898
  1320
proof (induct rule: finite_ne_induct)
haftmann@22917
  1321
  case singleton thus ?case by simp
haftmann@22917
  1322
next
haftmann@22917
  1323
  case (insert n N)
haftmann@26041
  1324
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1325
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1326
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1327
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1328
    using insert by(simp)
haftmann@22917
  1329
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1330
  finally show ?case .
haftmann@22917
  1331
qed
haftmann@22917
  1332
haftmann@32679
  1333
lemma fold1_eq_fold_idem:
haftmann@32679
  1334
  assumes "finite A"
haftmann@32679
  1335
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1336
proof (cases "a \<in> A")
haftmann@32679
  1337
  case False
haftmann@32679
  1338
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1339
next
haftmann@42871
  1340
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32679
  1341
  case True then obtain b B
haftmann@32679
  1342
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1343
  with assms have "finite B" by auto
haftmann@32679
  1344
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1345
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1346
  then show ?thesis
haftmann@32679
  1347
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1348
qed
haftmann@32679
  1349
haftmann@26041
  1350
end
haftmann@26041
  1351
paulson@15506
  1352
paulson@15508
  1353
text{* Now the recursion rules for definitions: *}
paulson@15508
  1354
haftmann@22917
  1355
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1356
by simp
paulson@15508
  1357
haftmann@26041
  1358
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1359
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1360
by (simp add:fold1_insert)
haftmann@26041
  1361
haftmann@26041
  1362
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1363
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1364
by simp
paulson@15508
  1365
paulson@15508
  1366
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1367
nipkow@28853
  1368
(*Not actually used!!*)
nipkow@28853
  1369
(*
haftmann@26041
  1370
context ab_semigroup_mult
haftmann@26041
  1371
begin
haftmann@26041
  1372
nipkow@28853
  1373
lemma fold_graph_permute:
nipkow@28853
  1374
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1375
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1376
apply (cases "a=b") 
nipkow@28853
  1377
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1378
done
nipkow@15376
  1379
haftmann@26041
  1380
lemma fold1Set_determ:
haftmann@26041
  1381
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1382
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1383
  fix A x B y a b
nipkow@28853
  1384
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1385
  assume By: "fold_graph times id b B y"
paulson@15506
  1386
  assume anotA:  "a \<notin> A"
paulson@15506
  1387
  assume bnotB:  "b \<notin> B"
paulson@15506
  1388
  assume eq: "insert a A = insert b B"
paulson@15506
  1389
  show "y=x"
paulson@15506
  1390
  proof cases
paulson@15506
  1391
    assume same: "a=b"
paulson@15506
  1392
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1393
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1394
  next
paulson@15506
  1395
    assume diff: "a\<noteq>b"
paulson@15506
  1396
    let ?D = "B - {a}"
paulson@15506
  1397
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1398
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1399
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1400
    with aB bnotB By
nipkow@28853
  1401
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1402
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1403
    moreover
nipkow@28853
  1404
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1405
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1406
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1407
  qed
wenzelm@12396
  1408
qed
wenzelm@12396
  1409
haftmann@26041
  1410
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1411
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1412
haftmann@26041
  1413
end
nipkow@28853
  1414
*)
haftmann@26041
  1415
paulson@15506
  1416
declare
nipkow@28853
  1417
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1418
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1419
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1420
haftmann@26041
  1421
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1422
haftmann@26041
  1423
context ab_semigroup_mult
haftmann@22917
  1424
begin
haftmann@22917
  1425
haftmann@26041
  1426
lemma fold1_Un:
nipkow@15484
  1427
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1428
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1429
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1430
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1431
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1432
haftmann@26041
  1433
lemma fold1_in:
haftmann@26041
  1434
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1435
  shows "fold1 times A \<in> A"
nipkow@15484
  1436
using A
nipkow@15484
  1437
proof (induct rule:finite_ne_induct)
paulson@15506
  1438
  case singleton thus ?case by simp
nipkow@15484
  1439
next
nipkow@15484
  1440
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1441
qed
nipkow@15484
  1442
haftmann@26041
  1443
end
haftmann@26041
  1444
haftmann@26041
  1445
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1446
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1447
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1448
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1449
using A
haftmann@26041
  1450
proof(induct rule:finite_ne_induct)
nipkow@15497
  1451
  case singleton thus ?case by simp
nipkow@15484
  1452
next
haftmann@26041
  1453
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1454
qed
nipkow@18423
  1455
nipkow@18423
  1456
haftmann@35817
  1457
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1458
haftmann@35817
  1459
subsubsection {* The natural case *}
haftmann@35719
  1460
haftmann@35719
  1461
locale folding =
haftmann@35719
  1462
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1463
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
  1464
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1465
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1466
begin
haftmann@35719
  1467
haftmann@35719
  1468
lemma empty [simp]:
haftmann@35719
  1469
  "F {} = id"
nipkow@39302
  1470
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1471
haftmann@35719
  1472
lemma insert [simp]:
haftmann@35719
  1473
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1474
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1475
proof -
wenzelm@46898
  1476
  interpret comp_fun_commute f
wenzelm@46898
  1477
    by default (insert comp_fun_commute, simp add: fun_eq_iff)
haftmann@35719
  1478
  from fold_insert2 assms
haftmann@35722
  1479
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1480
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1481
qed
haftmann@35719
  1482
haftmann@35719
  1483
lemma remove:
haftmann@35719
  1484
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1485
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1486
proof -
haftmann@35719
  1487
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1488
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1489
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1490
  ultimately show ?thesis by simp
haftmann@35719
  1491
qed
haftmann@35719
  1492
haftmann@35719
  1493
lemma insert_remove:
haftmann@35719
  1494
  assumes "finite A"
haftmann@35719
  1495
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1496
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1497
haftmann@35817
  1498
lemma commute_left_comp:
haftmann@35817
  1499
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@42871
  1500
  by (simp add: o_assoc comp_fun_commute)
haftmann@35817
  1501
haftmann@42871
  1502
lemma comp_fun_commute':
haftmann@35719
  1503
  assumes "finite A"
haftmann@35719
  1504
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1505
  using assms by (induct A)
haftmann@49739
  1506
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: comp_assoc comp_fun_commute)
haftmann@35817
  1507
haftmann@35817
  1508
lemma commute_left_comp':
haftmann@35817
  1509
  assumes "finite A"
haftmann@35817
  1510
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@42871
  1511
  using assms by (simp add: o_assoc comp_fun_commute')
haftmann@35817
  1512
haftmann@42871
  1513
lemma comp_fun_commute'':
haftmann@35817
  1514
  assumes "finite A" and "finite B"
haftmann@35817
  1515
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1516
  using assms by (induct A)
haftmann@49739
  1517
    (simp_all add: o_assoc, simp add: comp_assoc comp_fun_commute')
haftmann@35719
  1518
haftmann@35817
  1519
lemma commute_left_comp'':
haftmann@35817
  1520
  assumes "finite A" and "finite B"
haftmann@35817
  1521
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@42871
  1522
  using assms by (simp add: o_assoc comp_fun_commute'')
haftmann@35817
  1523
haftmann@49739
  1524
lemmas comp_fun_commutes = comp_assoc comp_fun_commute commute_left_comp
haftmann@42871
  1525
  comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
haftmann@35817
  1526
haftmann@35817
  1527
lemma union_inter:
haftmann@35817
  1528
  assumes "finite A" and "finite B"
haftmann@35817
  1529
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1530
  using assms by (induct A)
haftmann@42871
  1531
    (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
haftmann@35817
  1532
      simp add: o_assoc)
haftmann@35719
  1533
haftmann@35719
  1534
lemma union:
haftmann@35719
  1535
  assumes "finite A" and "finite B"
haftmann@35719
  1536
  and "A \<inter> B = {}"
haftmann@35719
  1537
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1538
proof -
haftmann@35817
  1539
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1540
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1541
qed
haftmann@35719
  1542
haftmann@34007
  1543
end
haftmann@35719
  1544
haftmann@35817
  1545
haftmann@35817
  1546
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1547
haftmann@35719
  1548
locale folding_idem = folding +
haftmann@35719
  1549
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1550
begin
haftmann@35719
  1551
haftmann@35817
  1552
lemma idem_left_comp:
haftmann@35817
  1553
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1554
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1555
haftmann@35817
  1556
lemma in_comp_idem:
haftmann@35817
  1557
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1558
  shows "F A \<circ> f x = F A"
haftmann@35817
  1559
using assms by (induct A)
haftmann@42871
  1560
  (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
haftmann@35719
  1561
haftmann@35817
  1562
lemma subset_comp_idem:
haftmann@35817
  1563
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1564
  shows "F A \<circ> F B = F A"
haftmann@35817
  1565
proof -
haftmann@35817
  1566
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1567
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1568
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1569
qed
haftmann@35719
  1570
haftmann@35817
  1571
declare insert [simp del]
haftmann@35719
  1572
haftmann@35719
  1573
lemma insert_idem [simp]:
haftmann@35719
  1574
  assumes "finite A"
haftmann@35719
  1575
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1576
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1577
haftmann@35719
  1578
lemma union_idem:
haftmann@35719
  1579
  assumes "finite A" and "finite B"
haftmann@35719
  1580
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1581
proof -
haftmann@35817
  1582
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1583
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1584
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1585
qed
haftmann@35719
  1586
haftmann@35719
  1587
end
haftmann@35719
  1588
haftmann@35817
  1589
haftmann@35817
  1590
subsubsection {* The image case with fixed function *}
haftmann@35817
  1591
haftmann@35796
  1592
no_notation times (infixl "*" 70)
haftmann@35796
  1593
no_notation Groups.one ("1")
haftmann@35722
  1594
haftmann@35722
  1595
locale folding_image_simple = comm_monoid +
haftmann@35722
  1596
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1597
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1598
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1599
begin
haftmann@35722
  1600
haftmann@35722
  1601
lemma empty [simp]:
haftmann@35722
  1602
  "F {} = 1"
haftmann@35817
  1603
  by (simp add: eq_fold_g)
haftmann@35722
  1604
haftmann@35722
  1605
lemma insert [simp]:
haftmann@35722
  1606
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1607
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1608
proof -
wenzelm@46898
  1609
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
  1610
    by default (simp add: ac_simps fun_eq_iff)
wenzelm@46898
  1611
  from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1612
    by (simp add: fold_image_def)
haftmann@35817
  1613
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1614
qed
haftmann@35722
  1615
haftmann@35722
  1616
lemma remove:
haftmann@35722
  1617
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1618
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1619
proof -
haftmann@35722
  1620
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1621
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1622
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1623
  ultimately show ?thesis by simp
haftmann@35722
  1624
qed
haftmann@35722
  1625
haftmann@35722
  1626
lemma insert_remove:
haftmann@35722
  1627
  assumes "finite A"
haftmann@35722
  1628
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1629
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1630
haftmann@35817
  1631
lemma neutral:
haftmann@35817
  1632
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1633
  shows "F A = 1"
haftmann@35817
  1634
  using assms by (induct A) simp_all
haftmann@35817
  1635
haftmann@35722
  1636
lemma union_inter:
haftmann@35722
  1637
  assumes "finite A" and "finite B"
haftmann@35817
  1638
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1639
using assms proof (induct A)
haftmann@35722
  1640
  case empty then show ?case by simp
haftmann@35722
  1641
next
haftmann@35722
  1642
  case (insert x A) then show ?case
haftmann@35722
  1643
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1644
qed
haftmann@35722
  1645
haftmann@35817
  1646
corollary union_inter_neutral:
haftmann@35817
  1647
  assumes "finite A" and "finite B"
haftmann@35817
  1648
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1649
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1650
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1651
haftmann@35722
  1652
corollary union_disjoint:
haftmann@35722
  1653
  assumes "finite A" and "finite B"
haftmann@35722
  1654
  assumes "A \<inter> B = {}"
haftmann@35722
  1655
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1656
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1657
haftmann@35719
  1658
end
haftmann@35722
  1659
haftmann@35817
  1660
haftmann@35817
  1661
subsubsection {* The image case with flexible function *}
haftmann@35817
  1662
haftmann@35722
  1663
locale folding_image = comm_monoid +
haftmann@35722
  1664
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1665
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1666
haftmann@35722
  1667
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1668
qed (fact eq_fold)
haftmann@35722
  1669
haftmann@35722
  1670
context folding_image
haftmann@35722
  1671
begin
haftmann@35722
  1672
haftmann@35817
  1673
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1674
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1675
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1676
  using assms by (induct A) auto
haftmann@35722
  1677
haftmann@35722
  1678
lemma cong:
haftmann@35722
  1679
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1680
  shows "F g A = F h A"
haftmann@35722
  1681
proof -
haftmann@35722
  1682
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1683
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1684
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1685
  apply (subgoal_tac "finite C")
wenzelm@48125
  1686
  prefer 2 apply (blast dest: finite_subset [rotated])
haftmann@35722
  1687
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1688
  prefer 2 apply blast
haftmann@35722
  1689
  apply (erule ssubst)
haftmann@35722
  1690
  apply (drule spec)
haftmann@35722
  1691
  apply (erule (1) notE impE)
haftmann@35722
  1692
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1693
  done
haftmann@35722
  1694
  with assms show ?thesis by simp
haftmann@35722
  1695
qed
haftmann@35722
  1696
haftmann@35722
  1697
lemma UNION_disjoint:
haftmann@35722
  1698
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1699
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1700
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1701
apply (insert assms)
haftmann@41656
  1702
apply (induct rule: finite_induct)
haftmann@41656
  1703
apply simp
haftmann@41656
  1704
apply atomize
haftmann@35722
  1705
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1706
 prefer 2 apply blast
haftmann@35722
  1707
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1708
 prefer 2 apply blast
haftmann@35722
  1709
apply (simp add: union_disjoint)
haftmann@35722
  1710
done
haftmann@35722
  1711
haftmann@35722
  1712
lemma distrib:
haftmann@35722
  1713
  assumes "finite A"
haftmann@35722
  1714
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1715
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1716
haftmann@35722
  1717
lemma related: 
haftmann@35722
  1718
  assumes Re: "R 1 1" 
haftmann@35722
  1719
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1720
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1721
  shows "R (F h S) (F g S)"
haftmann@35722
  1722
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1723
haftmann@35722
  1724
lemma eq_general:
haftmann@35722
  1725
  assumes fS: "finite S"
haftmann@35722
  1726
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1727
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1728
  shows "F f1 S = F f2 S'"
haftmann@35722
  1729
proof-
haftmann@35722
  1730
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1731
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1732
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1733
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1734
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1735
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1736
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1737
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1738
    by blast
haftmann@35722
  1739
  finally show ?thesis ..
haftmann@35722
  1740
qed
haftmann@35722
  1741
haftmann@35722
  1742
lemma eq_general_inverses:
haftmann@35722
  1743
  assumes fS: "finite S" 
haftmann@35722
  1744
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1745
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
haftmann@35722
  1746
  shows "F j S = F g T"
haftmann@35722
  1747
  (* metis solves it, but not yet available here *)
haftmann@35722
  1748
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1749
  apply (rule ballI)
haftmann@35722
  1750
  apply (frule kh)
haftmann@35722
  1751
  apply (rule ex1I[])
haftmann@35722
  1752
  apply blast
haftmann@35722
  1753
  apply clarsimp
haftmann@35722
  1754
  apply (drule hk) apply simp
haftmann@35722
  1755
  apply (rule sym)
haftmann@35722
  1756
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1757
  apply (rule ballI)
haftmann@35722
  1758
  apply (drule hk)
haftmann@35722
  1759
  apply blast
haftmann@35722
  1760
  done
haftmann@35722
  1761
haftmann@35722
  1762
end
haftmann@35722
  1763
haftmann@35817
  1764
haftmann@35817
  1765
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1766
haftmann@35817
  1767
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1768
  assumes idem: "x * x = x"
haftmann@35817
  1769
wenzelm@49756
  1770
sublocale folding_image_simple_idem < semilattice: semilattice proof
haftmann@35817
  1771
qed (fact idem)
haftmann@35817
  1772
haftmann@35817
  1773
context folding_image_simple_idem
haftmann@35817
  1774
begin
haftmann@35817
  1775
haftmann@35817
  1776
lemma in_idem:
haftmann@35817
  1777
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1778
  shows "g x * F A = F A"
haftmann@35817
  1779
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1780
haftmann@35817
  1781
lemma subset_idem:
haftmann@35817
  1782
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1783
  shows "F B * F A = F A"
haftmann@35817
  1784
proof -
haftmann@35817
  1785
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1786
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1787
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1788
qed
haftmann@35817
  1789
haftmann@35817
  1790
declare insert [simp del]
haftmann@35817
  1791
haftmann@35817
  1792
lemma insert_idem [simp]:
haftmann@35817
  1793
  assumes "finite A"
haftmann@35817
  1794
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1795
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1796
haftmann@35817
  1797
lemma union_idem:
haftmann@35817
  1798
  assumes "finite A" and "finite B"
haftmann@35817
  1799
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1800
proof -
haftmann@35817
  1801
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1802
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1803
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1804
qed
haftmann@35817
  1805
haftmann@35817
  1806
end
haftmann@35817
  1807
haftmann@35817
  1808
haftmann@35817
  1809
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1810
haftmann@35817
  1811
locale folding_image_idem = folding_image +
haftmann@35817
  1812
  assumes idem: "x * x = x"
haftmann@35817
  1813
haftmann@35817
  1814
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1815
qed (fact idem)
haftmann@35817
  1816
haftmann@35817
  1817
haftmann@35817
  1818
subsubsection {* The neutral-less case *}
haftmann@35817
  1819
haftmann@35817
  1820
locale folding_one = abel_semigroup +
haftmann@35817
  1821
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1822
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1823
begin
haftmann@35817
  1824
haftmann@35817
  1825
lemma singleton [simp]:
haftmann@35817
  1826
  "F {x} = x"
haftmann@35817
  1827
  by (simp add: eq_fold)
haftmann@35817
  1828
haftmann@35817
  1829
lemma eq_fold':
haftmann@35817
  1830
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1831
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1832
proof -
wenzelm@46898
  1833
  interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1834
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1835
qed
haftmann@35817
  1836
haftmann@35817
  1837
lemma insert [simp]:
haftmann@36637
  1838
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1839
  shows "F (insert x A) = x * F A"
haftmann@36637
  1840
proof -
haftmann@36637
  1841
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1842
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1843
  with `finite A` have "finite B" by simp
haftmann@35817
  1844
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1845
  qed (simp_all add: fun_eq_iff ac_simps)
haftmann@42871
  1846
  from `finite B` fold.comp_fun_commute' [of B x]
haftmann@35817
  1847
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1848
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1849
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1850
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1851
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1852
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1853
qed
haftmann@35817
  1854
haftmann@35817
  1855
lemma remove:
haftmann@35817
  1856
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1857
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1858
proof -
haftmann@35817
  1859
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1860
  with assms show ?thesis by simp
haftmann@35817
  1861
qed
haftmann@35817
  1862
haftmann@35817
  1863
lemma insert_remove:
haftmann@35817
  1864
  assumes "finite A"
haftmann@35817
  1865
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1866
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1867
haftmann@35817
  1868
lemma union_disjoint:
haftmann@35817
  1869
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1870
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1871
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1872
haftmann@35817
  1873
lemma union_inter:
haftmann@35817
  1874
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1875
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1876
proof -
haftmann@35817
  1877
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1878
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1879
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1880
  next
haftmann@35817
  1881
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1882
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1883
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1884
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1885
    next
haftmann@35817
  1886
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1887
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1888
        by auto
haftmann@35817
  1889
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1890
    qed
haftmann@35817
  1891
  qed
haftmann@35817
  1892
qed
haftmann@35817
  1893
haftmann@35817
  1894
lemma closed:
haftmann@35817
  1895
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1896
  shows "F A \<in> A"
haftmann@35817
  1897
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1898
  case singleton then show ?case by simp
haftmann@35817
  1899
next
haftmann@35817
  1900
  case insert with elem show ?case by force
haftmann@35817
  1901
qed
haftmann@35817
  1902
haftmann@35817
  1903
end
haftmann@35817
  1904
haftmann@35817
  1905
haftmann@35817
  1906
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1907
haftmann@35817
  1908
locale folding_one_idem = folding_one +
haftmann@35817
  1909
  assumes idem: "x * x = x"
haftmann@35817
  1910
wenzelm@49756
  1911
sublocale folding_one_idem < semilattice: semilattice proof
haftmann@35817
  1912
qed (fact idem)
haftmann@35817
  1913
haftmann@35817
  1914
context folding_one_idem
haftmann@35817
  1915
begin
haftmann@35817
  1916
haftmann@35817
  1917
lemma in_idem:
haftmann@35817
  1918
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1919
  shows "x * F A = F A"
haftmann@35817
  1920
proof -
haftmann@35817
  1921
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1922
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1923
qed
haftmann@35817
  1924
haftmann@35817
  1925
lemma subset_idem:
haftmann@35817
  1926
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1927
  shows "F B * F A = F A"
haftmann@35817
  1928
proof -
haftmann@35817
  1929
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1930
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1931
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1932
qed
haftmann@35817
  1933
haftmann@35817
  1934
lemma eq_fold_idem':
haftmann@35817
  1935
  assumes "finite A"
haftmann@35817
  1936
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1937
proof -
wenzelm@46898
  1938
  interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1939
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1940
qed
haftmann@35817
  1941
haftmann@35817
  1942
lemma insert_idem [simp]:
haftmann@36637
  1943
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1944
  shows "F (insert x A) = x * F A"
haftmann@35817
  1945
proof (cases "x \<in> A")
haftmann@36637
  1946
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1947
next
haftmann@36637
  1948
  case True
haftmann@36637
  1949
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1950
qed
haftmann@35817
  1951
  
haftmann@35817
  1952
lemma union_idem:
haftmann@35817
  1953
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1954
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1955
proof (cases "A \<inter> B = {}")
haftmann@35817
  1956
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1957
next
haftmann@35817
  1958
  case False
haftmann@35817
  1959
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1960
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1961
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1962
qed
haftmann@35817
  1963
haftmann@35817
  1964
lemma hom_commute:
haftmann@35817
  1965
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1966
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1967
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1968
  case singleton thus ?case by simp
haftmann@35817
  1969
next
haftmann@35817
  1970
  case (insert n N)
haftmann@35817
  1971
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1972
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1973
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1974
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1975
    using insert by(simp)
haftmann@35817
  1976
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1977
  finally show ?case .
haftmann@35817
  1978
qed
haftmann@35817
  1979
haftmann@35817
  1980
end
haftmann@35817
  1981
haftmann@35796
  1982
notation times (infixl "*" 70)
haftmann@35796
  1983
notation Groups.one ("1")
haftmann@35722
  1984
haftmann@35722
  1985
haftmann@35722
  1986
subsection {* Finite cardinality *}
haftmann@35722
  1987
haftmann@35722
  1988
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1989
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1990
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1991
*}
haftmann@35722
  1992
haftmann@35722
  1993
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1994
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1995
haftmann@37770
  1996
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1997
qed (simp add: card_def)
haftmann@35722
  1998
haftmann@35722
  1999
lemma card_infinite [simp]:
haftmann@35722
  2000
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  2001
  by (simp add: card_def)
haftmann@35722
  2002
haftmann@35722
  2003
lemma card_empty:
haftmann@35722
  2004
  "card {} = 0"
haftmann@35722
  2005
  by (fact card.empty)
haftmann@35722
  2006
haftmann@35722
  2007
lemma card_insert_disjoint:
haftmann@35722
  2008
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  2009
  by simp
haftmann@35722
  2010
haftmann@35722
  2011
lemma card_insert_if:
haftmann@35722
  2012
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  2013
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  2014
haftmann@35722
  2015
lemma card_ge_0_finite:
haftmann@35722
  2016
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  2017
  by (rule ccontr) simp
haftmann@35722
  2018
blanchet@35828
  2019
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  2020
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  2021
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  2022
haftmann@35722
  2023
lemma finite_UNIV_card_ge_0:
haftmann@35722
  2024
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  2025
  by (rule ccontr) simp
haftmann@35722
  2026
haftmann@35722
  2027
lemma card_eq_0_iff:
haftmann@35722
  2028
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  2029
  by auto
haftmann@35722
  2030
haftmann@35722
  2031
lemma card_gt_0_iff:
haftmann@35722
  2032
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  2033
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  2034
haftmann@35722
  2035
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  2036
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  2037
apply(simp del:insert_Diff_single)
haftmann@35722
  2038
done
haftmann@35722
  2039
haftmann@35722
  2040
lemma card_Diff_singleton:
haftmann@35722
  2041
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  2042
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  2043
haftmann@35722
  2044
lemma card_Diff_singleton_if:
bulwahn@45166
  2045
  "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  2046
by (simp add: card_Diff_singleton)
haftmann@35722
  2047
haftmann@35722
  2048
lemma card_Diff_insert[simp]:
haftmann@35722
  2049
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  2050
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  2051
proof -
haftmann@35722
  2052
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  2053
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  2054
qed
haftmann@35722
  2055
haftmann@35722
  2056
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  2057
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2058
haftmann@35722
  2059
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  2060
by (simp add: card_insert_if)
haftmann@35722
  2061
nipkow@41987
  2062
lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
nipkow@41987
  2063
by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
nipkow@41987
  2064
nipkow@41988
  2065
lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
nipkow@41987
  2066
using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
nipkow@41987
  2067
haftmann@35722
  2068
lemma card_mono:
haftmann@35722
  2069
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  2070
  shows "card A \<le> card B"
haftmann@35722
  2071
proof -
haftmann@35722
  2072
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  2073
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  2074
    case empty then show ?case by simp
haftmann@35722
  2075
  next
haftmann@35722
  2076
    case (insert x A)
haftmann@35722
  2077
    then have "x \<in> B" by simp
haftmann@35722
  2078
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  2079
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  2080
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  2081
  qed
haftmann@35722
  2082
qed
haftmann@35722
  2083
haftmann@35722
  2084
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  2085
apply (induct rule: finite_induct)
haftmann@41656
  2086
apply simp
haftmann@41656
  2087
apply clarify
haftmann@35722
  2088
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  2089
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  2090
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  2091
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  2092
apply (case_tac "card A", auto)
haftmann@35722
  2093
done
haftmann@35722
  2094
haftmann@35722
  2095
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  2096
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  2097
apply (blast dest: card_seteq)
haftmann@35722
  2098
done
haftmann@35722
  2099
haftmann@35722
  2100
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  2101
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  2102
  by (fact card.union_inter [symmetric])
haftmann@35722
  2103
haftmann@35722
  2104
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  2105
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  2106
  by (fact card.union_disjoint)
haftmann@35722
  2107
haftmann@35722
  2108
lemma card_Diff_subset:
haftmann@35722
  2109
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  2110
  shows "card (A - B) = card A - card B"
haftmann@35722
  2111
proof (cases "finite A")
haftmann@35722
  2112
  case False with assms show ?thesis by simp
haftmann@35722
  2113
next
haftmann@35722
  2114
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  2115
qed
haftmann@35722
  2116
haftmann@35722
  2117
lemma card_Diff_subset_Int:
haftmann@35722
  2118
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  2119
proof -
haftmann@35722
  2120
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  2121
  thus ?thesis
haftmann@35722
  2122
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  2123
qed
haftmann@35722
  2124
nipkow@40716
  2125
lemma diff_card_le_card_Diff:
nipkow@40716
  2126
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  2127
proof-
nipkow@40716
  2128
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  2129
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  2130
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  2131
  finally show ?thesis .
nipkow@40716
  2132
qed
nipkow@40716
  2133
haftmann@35722
  2134
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  2135
apply (rule Suc_less_SucD)
haftmann@35722
  2136
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2137
done
haftmann@35722
  2138
haftmann@35722
  2139
lemma card_Diff2_less:
haftmann@35722
  2140
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  2141
apply (case_tac "x = y")
haftmann@35722
  2142
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  2143
apply (rule less_trans)
haftmann@35722
  2144
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  2145
done
haftmann@35722
  2146
haftmann@35722
  2147
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  2148
apply (case_tac "x : A")
haftmann@35722
  2149
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  2150
done
haftmann@35722
  2151
haftmann@35722
  2152
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  2153
by (erule psubsetI, blast)
haftmann@35722
  2154
haftmann@35722
  2155
lemma insert_partition:
haftmann@35722
  2156
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
haftmann@35722
  2157
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  2158
by auto
haftmann@35722
  2159
haftmann@35722
  2160
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  2161
  assumes fin: "finite A" 
urbanc@36079
  2162
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  2163
  shows "P A"
urbanc@36079
  2164
using fin
urbanc@36079
  2165
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  2166
  case (less A)
urbanc@36079
  2167
  have fin: "finite A" by fact
urbanc@36079
  2168
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  2169
  { fix B 
urbanc@36079
  2170
    assume asm: "B \<subset> A"
urbanc@36079
  2171
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  2172
    moreover
urbanc@36079
  2173
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  2174
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  2175
    ultimately 
urbanc@36079
  2176
    have "P B" using ih by simp
urbanc@36079
  2177
  }
urbanc@36079
  2178
  with fin show "P A" using major by blast
haftmann@35722
  2179
qed
haftmann@35722
  2180
haftmann@35722
  2181
text{* main cardinality theorem *}
haftmann@35722
  2182
lemma card_partition [rule_format]:
haftmann@35722
  2183
  "finite C ==>
haftmann@35722
  2184
     finite (\<Union> C) -->
haftmann@35722
  2185
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2186
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2187
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2188
apply (erule finite_induct, simp)
haftmann@35722
  2189
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2190
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2191
done
haftmann@35722
  2192
haftmann@35722
  2193
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2194
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2195
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2196
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2197
proof
haftmann@35722
  2198
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2199
  show "UNIV \<subseteq> A"
haftmann@35722
  2200
  proof
haftmann@35722
  2201
    fix x
haftmann@35722
  2202
    show "x \<in> A"
haftmann@35722
  2203
    proof (rule ccontr)
haftmann@35722
  2204
      assume "x \<notin> A"
haftmann@35722
  2205
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2206
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2207
      with card show False by simp
haftmann@35722
  2208
    qed
haftmann@35722
  2209
  qed
haftmann@35722
  2210
qed
haftmann@35722
  2211
haftmann@35722
  2212
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2213
haftmann@35722
  2214
lemma card_eq_SucD:
haftmann@35722
  2215
assumes "card A = Suc k"
haftmann@35722
  2216
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2217
proof -
haftmann@35722
  2218
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2219
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2220
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2221
  show ?thesis
haftmann@35722
  2222
  proof (intro exI conjI)
haftmann@35722
  2223
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2224
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2225
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@44890
  2226
      using assms b fin by(fastforce dest:mk_disjoint_insert)+
haftmann@35722
  2227
  qed
haftmann@35722
  2228
qed
haftmann@35722
  2229
haftmann@35722
  2230
lemma card_Suc_eq:
haftmann@35722
  2231
  "(card A = Suc k) =
haftmann@35722
  2232
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2233
apply(rule iffI)
haftmann@35722
  2234
 apply(erule card_eq_SucD)
haftmann@35722
  2235
apply(auto)
haftmann@35722
  2236
apply(subst card_insert)
haftmann@35722
  2237
 apply(auto intro:ccontr)
haftmann@35722
  2238
done
haftmann@35722
  2239
nipkow@44744
  2240
lemma card_le_Suc_iff: "finite A \<Longrightarrow>
nipkow@44744
  2241
  Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
nipkow@44890
  2242
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
nipkow@44744
  2243
  dest: subset_singletonD split: nat.splits if_splits)
nipkow@44744
  2244
haftmann@35722
  2245
lemma finite_fun_UNIVD2:
haftmann@35722
  2246
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2247
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2248
proof -
haftmann@46146
  2249
  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@46146
  2250
    by (rule finite_imageI)
haftmann@46146
  2251
  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@46146
  2252
    by (rule UNIV_eq_I) auto
haftmann@35722
  2253
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2254
qed
haftmann@35722
  2255
huffman@48063
  2256
lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
haftmann@35722
  2257
  unfolding UNIV_unit by simp
haftmann@35722
  2258
huffman@47210
  2259
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
huffman@47210
  2260
  unfolding UNIV_bool by simp
huffman@47210
  2261
haftmann@35722
  2262
haftmann@35722
  2263
subsubsection {* Cardinality of image *}
haftmann@35722
  2264
haftmann@35722
  2265
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  2266
apply (induct rule: finite_induct)
haftmann@35722
  2267
 apply simp
haftmann@35722
  2268
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2269
done
haftmann@35722
  2270
haftmann@35722
  2271
lemma card_image:
haftmann@35722
  2272
  assumes "inj_on f A"
haftmann@35722
  2273
  shows "card (f ` A) = card A"
haftmann@35722
  2274
proof (cases "finite A")
haftmann@35722
  2275
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2276
next
haftmann@35722
  2277
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2278
  with False show ?thesis by simp
haftmann@35722
  2279
qed
haftmann@35722
  2280
haftmann@35722
  2281
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2282
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2283
haftmann@35722
  2284
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2285
by (simp add: card_seteq card_image)
haftmann@35722
  2286
haftmann@35722
  2287
lemma eq_card_imp_inj_on:
haftmann@35722
  2288
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2289
apply (induct rule:finite_induct)
haftmann@35722
  2290
apply simp
haftmann@35722
  2291
apply(frule card_image_le[where f = f])
haftmann@35722
  2292
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2293
done
haftmann@35722
  2294
haftmann@35722
  2295
lemma inj_on_iff_eq_card:
haftmann@35722
  2296
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2297
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2298
haftmann@35722
  2299
haftmann@35722
  2300
lemma card_inj_on_le:
haftmann@35722
  2301
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2302
apply (subgoal_tac "finite A") 
haftmann@35722
  2303
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2304
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2305
done
haftmann@35722
  2306
haftmann@35722
  2307
lemma card_bij_eq:
haftmann@35722
  2308
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2309
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2310
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2311
hoelzl@40703
  2312
lemma bij_betw_finite:
hoelzl@40703
  2313
  assumes "bij_betw f A B"
hoelzl@40703
  2314
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2315
using assms unfolding bij_betw_def
hoelzl@40703
  2316
using finite_imageD[of f A] by auto
haftmann@35722
  2317
haftmann@41656
  2318
nipkow@37466
  2319
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2320
nipkow@40311
  2321
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2322
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2323
nipkow@37466
  2324
lemma pigeonhole_infinite:
nipkow@37466
  2325
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2326
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2327
proof -
nipkow@37466
  2328
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2329
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2330
    case empty thus ?case by simp
nipkow@37466
  2331
  next
nipkow@37466
  2332
    case (insert b F)
nipkow@37466
  2333
    show ?case
nipkow@37466
  2334
    proof cases
nipkow@37466
  2335
      assume "finite{a:A. f a = b}"
nipkow@37466
  2336
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2337
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2338
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2339
      from insert(3)[OF _ this]
nipkow@37466
  2340
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2341
    next
nipkow@37466
  2342
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2343
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2344
      thus ?thesis using 1 by blast
nipkow@37466
  2345
    qed
nipkow@37466
  2346
  qed
nipkow@37466
  2347
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2348
qed
nipkow@37466
  2349
nipkow@37466
  2350
lemma pigeonhole_infinite_rel:
nipkow@37466
  2351
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2352
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2353
proof -
nipkow@37466
  2354
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2355
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2356
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2357
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2358
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2359
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2360
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2361
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2362
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2363
   }
nipkow@37466
  2364
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2365
qed
nipkow@37466
  2366
nipkow@37466
  2367
haftmann@35722
  2368
subsubsection {* Cardinality of sums *}
haftmann@35722
  2369
haftmann@35722
  2370
lemma card_Plus:
haftmann@35722
  2371
  assumes "finite A" and "finite B"
haftmann@35722
  2372
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2373
proof -
haftmann@35722
  2374
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2375
  with assms show ?thesis
haftmann@35722
  2376
    unfolding Plus_def
haftmann@35722
  2377
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2378
qed
haftmann@35722
  2379
haftmann@35722
  2380
lemma card_Plus_conv_if:
haftmann@35722
  2381
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2382
  by (auto simp add: card_Plus)
haftmann@35722
  2383
haftmann@35722
  2384
haftmann@35722
  2385
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2386
huffman@47221
  2387
lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
haftmann@41656
  2388
apply (induct rule: finite_induct)
haftmann@35722
  2389
 apply (simp_all add: Pow_insert)
haftmann@35722
  2390
apply (subst card_Un_disjoint, blast)
nipkow@40786
  2391
  apply (blast, blast)
haftmann@35722
  2392
apply (subgoal_tac "inj_on (insert x) (Pow F)")
huffman@47221
  2393
 apply (subst mult_2)
haftmann@35722
  2394
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2395
apply (unfold inj_on_def)
haftmann@35722
  2396
apply (blast elim!: equalityE)
haftmann@35722
  2397
done
haftmann@35722
  2398
nipkow@41987
  2399
text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
haftmann@35722
  2400
haftmann@35722
  2401
lemma dvd_partition:
haftmann@35722
  2402
  "finite (Union C) ==>
haftmann@35722
  2403
    ALL c : C. k dvd card c ==>
haftmann@35722
  2404
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2405
  k dvd card (Union C)"
haftmann@41656
  2406
apply (frule finite_UnionD)
haftmann@41656
  2407
apply (rotate_tac -1)
haftmann@41656
  2408
apply (induct rule: finite_induct)
haftmann@41656
  2409
apply simp_all
haftmann@41656
  2410
apply clarify
haftmann@35722
  2411
apply (subst card_Un_disjoint)
haftmann@35722
  2412
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2413
done
haftmann@35722
  2414
haftmann@35722
  2415
haftmann@35722
  2416
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2417
haftmann@41656
  2418
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  2419
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2420
apply(frule finite_imageI)
haftmann@35722
  2421
apply(drule (1) card_seteq)
haftmann@35722
  2422
 apply(erule card_image_le)
haftmann@35722
  2423
apply simp
haftmann@35722
  2424
done
haftmann@35722
  2425
haftmann@35722
  2426
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2427
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2428
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2429
haftmann@35722
  2430
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2431
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@44890
  2432
by(fastforce simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2433
haftmann@35722
  2434
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2435
proof
haftmann@35722
  2436
  assume "finite(UNIV::nat set)"
haftmann@35722
  2437
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2438
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2439
qed
haftmann@35722
  2440
blanchet@35828
  2441
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2442
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2443
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2444
proof
haftmann@35722
  2445
  assume "finite (UNIV::'a set)"
haftmann@35722
  2446
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2447
    by (rule finite_subset)
haftmann@35722
  2448
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2449
    by (simp add: inj_on_def)
haftmann@35722
  2450
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2451
    by (rule finite_imageD)
haftmann@35722
  2452
  then show "False"
haftmann@35722
  2453
    by simp
haftmann@35722
  2454
qed
haftmann@35722
  2455
kuncar@49758
  2456
hide_const (open) Finite_Set.fold
haftmann@46033
  2457
haftmann@35722
  2458
end
haftmann@49723
  2459